Linear Key Predistribution Scheme

1. Linear Key Predistribution Scheme ShahzadBasiri Imam Hossein university shahzad_basiri@yahoo.com Workshop on key distribution Tuesday, May 24, 2011

2. Outline • Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality

3. KKey Distribution Scheme A key predistribution scheme (KPS) is a method by which • A trusted authority TA distributes secret information among a set of users in such a way that every user in a group in some specified family of privileged subsets is able to compute a common key associated with that group. • Besides, certain coalitions of users ( forbidden subsets) outside a privileged group must not be able to find out any information on the value of the key associated with that group.

4. Previous Scheme • Polynomial • Blom • Blundo • Based on Key Distribution Patterns • Mitchell and Piper • Trivial Scheme

5. Goals • One of the goals of this scheme is the construction of key predistribution schemes with good information rate for other families of specification structures. • A new general model for the design of key predistribution schemes, which is based mainly on linear algebraic techniques, the linear key predistribution schemes (LKPSs). • This new model, based on linear algebraic techniques, unifies all previous proposals.

7. Assumption • A subset P⊂Uis a privileged subset of the specification structure if there exists F⊂Usuch that (P, F) ∈ . • The family of the privileged subsets of is denoted by P( ). • For any P ∈ P( ),let us consider FP = {F⊂U : (P, F)∈ }. The elements of FP are called the P-forbidden subsets of .

8. Assumption • For any P ∈ P( ), the family of P-forbidden subsets FP is monotone decreasing, that is, if F1 ∈ FP and F2 ⊂ F1, then F2 ∈ FP . • For any F⊂U, we consider the family PF of F-privileged subsets of , which consists of all subsets P⊂U such that (P, F) ∈ .

9. Assumption • Let be a specification structure on a set of users U such that both FP and PFare monotone decreasing for any (P, F) ∈ . • The specification structure = {(P, F) ∈ : (F, P) ∈ } is called the dual specification structure of .

10. Outline • Key Predistribution Schemes • Linear Key Predistribution Schemes • Definition • Previous methods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality

13. Proof TA 1 2 3 Randomly chooses N

14. Outline • Key Predistribution Schemes • Linear Key Predistribution Schemes • Definition • Previous methods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality

15. Previous Scheme • Polynomial • Blom • Blundo • Based on Key Distribution Patterns • Mitchell and Piper • Trivial Scheme

17. Construct a KPS from KDP B1 TA B2 Randomly chooses Bl Bm

18. Construct a KPS from KDP P

20. Proof

21. Blundo et al scheme 1 TA 2 i ui = f (si , x2, . . . , xr ) N Randomly Choose f (x1, x2, . . . , xr ) Choose distinc public s1, s2, . . . , sl

22. Blundo et al scheme

23. Blundo et al scheme LKPS • Let Er be the vector space of symmetric polynomials on r variables, with coefficients inFq and degree at most t on each variable

24. Outline • Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality

26. Multilinear function • Let V be a vector space over a finite field Fq . • will denote the vector spaceV×V ×・・ ・×V, where there are r factors in this product. • A mapping T : Fq is called a multilinear function if, for any i = 1, 2, . . . , r , • T (v1, . . . , vi +v’i, . . . , vr ) = T (v1, . . . , vi, . . . , vr )+T (v1, . . . , v’i, . . . , vr ) • and • T (v1, . . . , λvi, . . . , vr ) = λT (v1, . . . , vi, . . . , vr ).

27. Notations • Notation 1: The vector space r -linear functions over Fq • Notation 2: The vector space r –linear symmetric functions over Fq

30. Outline • Key Predistribution Schemes • Linear Key Predistribution Schemes • Previous Metods in KPS • Constructing Linear Key Predistribution Schemes • Dulity in Linear Key Predistribution Schemes • Constructing Linear Key Predistribution Schemes by Duality

31. Duality in LKPSs Under certain conditions, any -LKPS provides a LKPS for the dual specification structure and we relate the information rates of the two schemes.

35. Proof (P, F) ∈ (F, P) ∈ There exists a -LKPS with information rate

37. ∈U